Georg Cantor

Georg Cantor was a German mathematician who founded set theory and developed a rigorous theory of infinite cardinalities, showing that infinities can have different sizes.

Set theory: The mathematical study of sets, their elements, and relations between collections. Cantor established this field as a foundation for studying mathematical infinity.
Cantor's diagonal argument: A proof technique that constructs an item differing from every item in a proposed list at a selected position. Cantor used it to prove that real numbers cannot be listed in a sequence.
University of Halle: A German university where Cantor spent most of his academic career. Cantor held his professorship there while developing set theory.
Zermelo–Fraenkel set theory: An axiomatic foundation for mathematics built around sets and rules governing their existence. Modern set theory formalizes the mathematical universe Cantor helped create.
Bijection: A function that pairs every element of one set with exactly one element of another, with none left over. Cantor used one-to-one pairings to define when two sets have the same size.
Cantor's theorem: The theorem that the power set of any set has strictly greater cardinality than the set itself. It establishes an endless hierarchy of larger infinite cardinalities.
Eduard Heine: A German mathematician whose work on Fourier series prompted Cantor’s early research into trigonometric series. Heine’s uniqueness problem helped lead Cantor toward the theory of infinite sets.
Hilbert's first problem: David Hilbert’s challenge to determine whether the continuum hypothesis is true. Cantor’s unresolved question became the first problem in Hilbert’s 1900 list.
Countable set: A set that is finite or whose elements can be listed in a sequence indexed by natural numbers. Cantor showed that some infinite sets can be listed, while others cannot.
Power set: The set of all subsets of a given set. Cantor’s theorem compares a set’s size with the size of its power set.

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